I lived in Hamburg for a year.
rent is very high, at least for my standards. we lived in a crappy place with airplane noise, 2 rooms (plus kitchen and bath) and the bus to the city center took over 30 minutes and we still paid about 550€ for it. the popular areas like St. Pauli, Altona to some extent, Eimsbüttel are also expensive, but we also only had one income. you might be able to afford it. you can check www.abendblatt.de
for "immobilien". there is a lot of competition for apartments, too, so that might be difficult. not trying to scare you, it might just take some time, so you'd better start looking early.
Hamburg is very international and while cashiers at the grocery stores will probably not speak English the public library has a vast English book section and you should be able to navigate the city easily. I think there is a website somewhere about an English-speaking community... there is also an English Theatre, book store and also an American Club or something? not sure on that as I am German. some info here
you definitely do not need a car. there's regular buses and the subway system, also s-bahn for longer distances. you can get monthly passes for those. you can also bike a lot.
I don't have children but I know there's children's book stores, plenty of playgrounds, the Elbe beach, a zoo... some moms might have more info on that for you. there are enough tourity things to keep you busy and a lively scene.
Hamburg is full of things to do but it is a pretty expensive and large town. I fel that it was great if you had the financial means to experience all it has to offer